Challenge Level

Think about what stays the same and what changes if you fix $c$ and vary $a+b$.

For the first part you have a right-angled triangle so you might try to use Pythagoras' Theorem which suggests working with $(a+b)^2$. The key idea is to maximise the area of the triangle.

To generalise this go from Pythagoras' Theorem to the Cosine Rule. Again use the area of the triangle. The key idea here is that if you fix $c$ and vary $a+b$ the angle $\angle ACB$ is constant.

The last part calls for a careful argument based on four applications of the second result.